An m  n matrix is an rectangular array of elements with m rows and n columns: Matrices denotes the element in the ith row and jth column

Partitioning in submatrices

Matrices y vectores son fundamentales en el estudio formal de todas las ramas de la ingeniería Instrumentación Instrumentación Diseño de circuitos Diseño de circuitos Comunicaciones Comunicaciones Microelectrónica Microelectrónica

A column vector is a matrix with n rows and 1 column Vectors A row vector is a matrix with 1 row and n columns

Square: Classification of matrices m=n

Symmetric: a ji = a ij

Upper Triangular: a ij = 0 when j < i

Lower Triangular: a ij = 0 when j >i

Diagonal: a ij = 0 when j  i

Identity: a ii = 1 a ij = 0 when j  i

Sum of matrices of the same dimension:

Scalar multiplication B = kA B = kA Dimensions: Dimensions: Example Example

Matrix multiplication C = AB C = AB Only possible if the number of columns of A is equal to the number of rows of B

examples:

Matrix multiplication is a non-commutative operation : Matrix multiplication is a non-commutative operation (generally) :

Identity: a ii = 1 a ij = 0 when j  i

Vector products: (u,v are column vectors) Dot product or inner product Dot product or inner product Outer product: Outer product:

Scalar product (of vectors) The product of a row vector a and a column vector b is a scalar a  b = a 1 b a n b n

Trace The trace of a nxn matrix A is given by:

Properties of Matrix Operations a) A+B = B+A b) A+(B+C) = (A+B)+C c) A(BC) = (AB)C d) A(B+C) = AB+AC e) (B+C)A = BA+CA f) a(B+C) = aB+aC Commutative law for addition Associative law for addition Associative for multiplication Left distributive law Right distributive law Distributive law for scalar multiplication

j) (a+b)C = aC+bC k) a(bC) = (ab)C l) a(BC) = (aB)C

Transpose B = A T Dimensions: Dimensions: Formula: Formula: Example Example

Alternative notation used in some books B = A T B = A ’ In this course we use the first one (B = A T )

Transpose Matrix properties

Symmetric matrix: A T = A Symmetric matrix: A T = A Skew-symmetric matrix: A T = -A Skew-symmetric matrix: A T = -A

Unitary matrix example : Unitary matrix example :

SymmetricSkew-symmetric Unitary matrix

Given any matrix A with real entries: Given any matrix A with real entries:

Complex conjugate of matrices

Alternative notation used in some books for Matrix Complex Conjugate In this notes we use the bar

Complex Hermitian Example:

Complex Hermitian Properties

definitions

examples: examples: Hermitian: Skew-Hermitian Unitary

Given any matrix A with complex entries: Given any matrix A with complex entries:

(a) Find A such as: (b) Find A such as: Exercises :

Exercises

Ejercicio: Simplificar Ejercicio: Simplificar